Uniform Continuity vs. Continuity

We have discussed some very useful properties of continuous functions in the last few posts. In the current post we will focus on another property called "uniform continuity". To understand what this is all about it first makes sense to reiterate the meaning of usual concept of continuity as we have seen earlier.

After discussing the boundedness property of the continuous functions, its time now to discuss another fundamental property of continuous functions called the Intermediate Value Theorem. This roughly means that if a continuous function takes two values $ A$ and $ B$ then it takes all values between $ A$ and $ B$. Thus the values of the function also maintain a continuity starting from one value to another. This theorem has some practical applications in solving equations for example. But more than that this is the most widely advertized property of continuous functions and is mentioned in almost every calculus book (and normally without proof). Some authors contend that this is the essence of continuity. However it is not the case as there can be discontinuous functions which possess this property.

In the previous post we had developed the concept of continuous functions and some of their local properties. It is now time to study some of the properties which apply to functions which are continuous on an interval. It turns out that the most useful and beautiful results present themselves when we study the functions defined on a closed interval. The magic goes away when the intervals under discussion are not closed. Why this happens (i.e. no magical properties for open intervals) is a further subtle point which we will not discuss rightaway.

Continuous Curves

In the last two posts we discussed about the system of real numbers and understood how the real numbers provide a model for the geometrical notion of a line which is continuous without any gaps. In this post and a few forthcoming ones we will try to create a model for continuous curves. Readers are already familiar with various continuous curves for example, circle, parabola and conic sections in general. The intuitive notion about the continuity of a curve is that we can draw in on a paper without lifting pen. This fact is so intuitive that many readers will feel that it is too much of an overkill to model it using some mathematical construct.

In the last post we wanted to create a system of numbers which have the following property:If all the numbers be divided into two sets $ L, U$ such that every member of $ L$ is less than any member of $ U$ then there must be a number $ \alpha$ such that all numbers less than $ \alpha$ belong to $ L$ and all the numbers greater than $ \alpha$ belong to $ U$, the number $ \alpha$ may itself lie in one of the sets $ L, U$.

Prologue

I remember a dialogue I had with a member of the Mathematics community on Orkut about the concept of real numbers. This was around 4-5 years ago and he was one of the really intelligent members out there in that community and had just joined IIT Bombay. To protect his identity lets call him by the name X. So we have a discussion which goes something like the following:

Me: So you have been doing great solving the math problems posed by the community.X: Sort of. I really love solving problems.Me: Looking at the depth of your skills, I suppose you must have some idea of real numbers.X: Oh yes, they are just below the Complex Numbers in the number system hierarchy.Me: Yeah that's fine, but what would you tell about real numbers to a guy who does not know about the complex numbers?X: I would say that the real numbers are "the rational numbers and the irrational numbers" taken together.Me: And then what would say about the irrational numbers?X: Irrationals are just those real numbers which are not rational.Me: If you look carefully at what you said, you will notice that there is a circularity involved and you have not defined any of the terms "irrational" and "real" in context of numbers.X: Yes I guess that you are correct, but I don't know how we can avoid that circularity. We have never been told otherwise about real numbers.